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H^{E} =

kJ/mol

ΔT =

K

S^{E} =

J/(mol K)

This step-by-step simulation concerns non-ideal mixing and partial molar quantities. The simulation randomly selects between entropy or enthalpy, and the user determines excess entropy or enthalpy, temperature change of mixing, and partial molar entropy or enthalpy. Directions for each step are given at the top of the plot. After completing each step, the user selects "show solution" to see the answer (green button above the plot). At any time, the user can select "new problem" to start over; it switches between entropy and enthalpy every time it starts over. The properties of components A and B (pure-component enthalpy/entropy, deviation from ideal behavior, etc.) are randomized every time "new problem" is selected.

The molar enthalpy \( H \) is given by:

$$ [1] \quad H^{E} = x_{A} H_{A} + x_{B} H_{B} + \alpha x_{A} x_{B} $$

where \( H_{A} \) and \( H_{B} \) are the pure-component enthalpies (kJ/mol), \( x_{A} \) and \( x_{B} \) are the mole fractions of \( A \) and \( B \) and \( \alpha \) is a non-ideal parameter, which can be either a positive or negative value. The excess enthalpy \( H^{E} \) is:

$$ [2] \quad H^{E} = H - H^{IS} = \alpha \, x_{A} \, x_{B} $$

where \( H^{IS} \) is the enthalpy of an ideal solution. The change in temperature for adiabatic mixing \( \Delta T_{mix} \) is a function of the excess enthalpy \( H^{E} \) and the heat capacity of the solution (\( C_{p} \)).

$$ [3] \quad H^{E} = - C_{p} \, \Delta T_{mix} = - C_{p} (T_{f} - T_{i}) $$

where \( T_{i} \) and \( T_{f} \) represent the initial and final temperatures of the solution. The partial molar enthalpies for components \( A \) and \( B \), \( \; \overline{H}_{A} \) and \( \overline{H}_{B} \), are at the intersections at \( x_{A} = 1 \) and \( x_{B} = 1 \), respectively, of a line tangent to the \( H \) versus \( x_{A} \) curve at the mixture mole fraction \( x_{A} \):

$$ [4] \quad \overline{H}_{A} = H + x_{B} \frac{dH}{dx_{A}} $$ $$ [5] \quad \overline{H}_{B} = H - x_{A} \frac{dH}{dx_{A}} $$

The molar entropy of an ideal solution \( S^{IS} \) is:

$$ [6] \quad S^{IS} = (x_{A} S_{A} + x_{B} S_{B}) - R (x_{A} \, \mathrm{ln} (x_{A}) + x_{B} \, \mathrm{ln} (x_{B})) $$

where \( R \) is the ideal gas constant. The excess entropy \( S^{E} \) is

$$ [7] \quad S^{E} = S - S^{IS} $$

The partial molar entropies can be found from interactions of a line tangent to the \( S \) versus \( x_{A} \) curve using equations analogous to equations [4] and [5].

This simulation was created in the Department of Chemical and Biological Engineering, at University of Colorado Boulder for LearnChemE.com by Neil Hendren under the direction of Professor John L. Falconer. This simulation was prepared with financial support from the National Science Foundation. Address any questions or comments to learncheme@gmail.com.

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